IB Math AA SL Topic 1.1: Standard form | Notes & Flashcards | Aimnova

IB Math AA SL — Standard form

Topic 1.1 of IB Mathematics: Analysis and Approaches covers Standard form, which is part of Unit 1: Number & Algebra. Students explore key concepts including Writing standard form, Standard form by hand. A strong understanding of standard form is essential for IB Math AA SL exams and builds the foundation for connected topics across the syllabus.

Key concepts in Standard form

Key Idea: Standard form is the IB's way of writing very large and very small numbers compactly. It usually appears as the last line of a question — give your answer in the form a × 10ᵏ — on both papers.

🔢 The structure: a × 10ᵏ

a \times 10^{k}

$1 \le a < 10$: the coefficient — one non-zero digit before the decimal point
$k \in \mathbb{Z}$: the exponent — a positive or negative whole number

Big number (≥ 10) → exponent positive, point moves left. e.g. 52 000 = 5.2 × 10⁴. Small number (< 1) → exponent negative, point moves right. e.g. 0.0007 = 7 × 10⁻⁴. Count place moves, not digits — and check 1 ≤ a < 10 every time.

🖐️ Calculating by hand (Paper 1)

✏️ IB-style worked examples

IB-style question — write a small number in standard form

Write 0.000 805 in standard form.

Step by step:

Put the point just after the first non-zero digit (8).
$8.05$
The point moved 4 places right, so the exponent is negative.
$k = -4$
Write them together.
$8.05 \times 10^{-4}$

Final answer:

0.000 805 = 8.05 × 10⁻⁴

IB-style question — compute, then express (Paper 2)

A sphere has radius 9.4 cm. Find its volume in the form a × 10ᵏ cm³, where 1 ≤ a < 10.

Step by step:

Use the volume formula and your GDC to work out the value.
$V = \tfrac{4}{3}\pi r^{3} = \tfrac{4}{3}\pi (9.4)^{3}$
The GDC does the arithmetic.
$V = 3479.14\ldots$
Rewrite in standard form — move the point 3 places.
$V = 3.48 \times 10^{3}$

Final answer:

V = 3.48 × 10³ cm³ (3 s.f.)

IB-style question — cube and re-normalise (Paper 1)

A cube has edge length 4 × 10² cm. Find its volume in the form a × 10ᵏ cm³, where 1 ≤ a < 10, without a calculator.

Step by step:

Volume of a cube = edge³. Cube the coefficient and multiply the exponent.
$V = (4 \times 10^{2})^{3} = 4^{3} \times 10^{2 \times 3}$
Work each part out.
$= 64 \times 10^{6}$
64 is not between 1 and 10 — re-normalise: 64 = 6.4 × 10¹.
$= 6.4 \times 10^{7}$

Final answer:

V = 6.4 × 10⁷ cm³

Important: The final answer must have exactly one non-zero digit before the decimal point (1 ≤ a < 10). After a calculation, re-normalise: 27 × 10⁶ → 2.7 × 10⁷, 0.5 × 10⁻³ → 5 × 10⁻⁴. With powers, raise the coefficient too: (4 × 10²)³ = 64 × 10⁶, not 4 × 10⁶.

Tap each card to reveal the answer.

Exam Tips

Check the coefficient a is between 1 and 10 — every time, including after a calculation.
Big number → positive exponent; small number → negative exponent. Count place moves, not digits.
Multiply → add powers. Divide → subtract. Power → multiply (and raise the coefficient too).
Re-normalise an untidy answer: 27 × 10⁶ → 2.7 × 10⁷.
On Paper 2, the GDC's ᴇ means × 10 — rewrite it; never leave ᴇ in your final answer.

IB Math AA SL — Standard form

Key concepts in Standard form

Key Idea: Standard form is the IB's way of writing very large and very small numbers compactly. It usually appears as the last line of a question — give your answer in the form a × 10ᵏ — on both papers.

🔢 The structure: a × 10ᵏ

a \times 10^{k}

$1 \le a < 10$: the coefficient — one non-zero digit before the decimal point
$k \in \mathbb{Z}$: the exponent — a positive or negative whole number

Big number (≥ 10) → exponent positive, point moves left. e.g. 52 000 = 5.2 × 10⁴. Small number (< 1) → exponent negative, point moves right. e.g. 0.0007 = 7 × 10⁻⁴. Count place moves, not digits — and check 1 ≤ a < 10 every time.

🖐️ Calculating by hand (Paper 1)

✏️ IB-style worked examples

IB-style question — write a small number in standard form

Write 0.000 805 in standard form.

Step by step:

Put the point just after the first non-zero digit (8).
$8.05$
The point moved 4 places right, so the exponent is negative.
$k = -4$
Write them together.
$8.05 \times 10^{-4}$

Final answer:

0.000 805 = 8.05 × 10⁻⁴

IB-style question — compute, then express (Paper 2)

A sphere has radius 9.4 cm. Find its volume in the form a × 10ᵏ cm³, where 1 ≤ a < 10.

Step by step:

Use the volume formula and your GDC to work out the value.
$V = \tfrac{4}{3}\pi r^{3} = \tfrac{4}{3}\pi (9.4)^{3}$
The GDC does the arithmetic.
$V = 3479.14\ldots$
Rewrite in standard form — move the point 3 places.
$V = 3.48 \times 10^{3}$

Final answer:

V = 3.48 × 10³ cm³ (3 s.f.)

IB-style question — cube and re-normalise (Paper 1)

A cube has edge length 4 × 10² cm. Find its volume in the form a × 10ᵏ cm³, where 1 ≤ a < 10, without a calculator.

Step by step:

Volume of a cube = edge³. Cube the coefficient and multiply the exponent.
$V = (4 \times 10^{2})^{3} = 4^{3} \times 10^{2 \times 3}$
Work each part out.
$= 64 \times 10^{6}$
64 is not between 1 and 10 — re-normalise: 64 = 6.4 × 10¹.
$= 6.4 \times 10^{7}$

Final answer:

V = 6.4 × 10⁷ cm³

Important: The final answer must have exactly one non-zero digit before the decimal point (1 ≤ a < 10). After a calculation, re-normalise: 27 × 10⁶ → 2.7 × 10⁷, 0.5 × 10⁻³ → 5 × 10⁻⁴. With powers, raise the coefficient too: (4 × 10²)³ = 64 × 10⁶, not 4 × 10⁶.

Tap each card to reveal the answer.

Exam Tips

Check the coefficient a is between 1 and 10 — every time, including after a calculation.
Big number → positive exponent; small number → negative exponent. Count place moves, not digits.
Multiply → add powers. Divide → subtract. Power → multiply (and raise the coefficient too).
Re-normalise an untidy answer: 27 × 10⁶ → 2.7 × 10⁷.
On Paper 2, the GDC's ᴇ means × 10 — rewrite it; never leave ᴇ in your final answer.

IB Math AA SL — Standard form

Key concepts in Standard form

🔢 The structure: a × 10ᵏ

🖐️ Calculating by hand (Paper 1)

✏️ IB-style worked examples

IB-style question — write a small number in standard form

IB-style question — compute, then express (Paper 2)

IB-style question — cube and re-normalise (Paper 1)

Exam Tips

What you'll learn in Topic 1.1

Study resources — 1.1 Standard form

Writing standard form

Standard form by hand

Ready to study Standard form?

Ready to practice?

IB Math AA SL — Standard form

Key concepts in Standard form

🔢 The structure: a × 10ᵏ

🖐️ Calculating by hand (Paper 1)

✏️ IB-style worked examples

IB-style question — write a small number in standard form

IB-style question — compute, then express (Paper 2)

IB-style question — cube and re-normalise (Paper 1)

Exam Tips

What you'll learn in Topic 1.1

Study resources — 1.1 Standard form

Writing standard form

Standard form by hand

Ready to study Standard form?

Ready to practice?